Journal fu¨r die reine und angewandte Mathematik
J. reine angew. Math. 611 (2007), 75—108 DOI 10.1515/CRELLE.2007.074
( Walter de Gruyter Berlin New York 2007
On the first and second variations of a nonlocal isoperimetric problem By Rustum Choksi1) at Burnaby and Peter Sternberg2) at Bloomington
Abstract. We consider a nonlocal perturbation of an isoperimetric variational problem. The problem may be viewed as a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation associated with competing short and long-range interactions. In particular, it arises as a G-limit of a model for microphase separation of diblock copolymers. In this article, we establish precise conditions for criticality and stability (i.e. we explicitly compute the first and second variations). We also present some applications.
1. Introduction This article is devoted to the computation and application of formulas for the first and second variation of the following nonlocal variational problem: For fixed m A ð1; 1Þ, ðNLIPÞ
minimize Eg ðuÞ :¼
Ð 1Ð j‘uj þ g j‘vj 2 dx; 2W W
over all u A BV ðW; fG1gÞ
with
1 Ð u dx ¼ m jWj W
and hv ¼ u m
in W:
We consider both the periodic case, where W ¼ T n , the n-dimensional flat torus of unit volume, and the homogeneous Neumann case where W is a general bounded domain in R n . In the former, the di¤erential equation is solved over the torus T n , and in the latter we impose Neumann boundary conditions (i.e. ‘v n ¼ 0 on qW). The first integral in (NLIP) represents the total variation of u, which for a function taking only values G1 is simply the pe-
1) Research partially supported by an NSERC (Canada) Discovery Grant. 2) Research partially supported by NSF DMS-0401328 and by the Institute for Mathematics and its Applications at the University of Minnesota.
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Choksi and Sternberg, Nonlocal isoperimetric problem
rimeter of the set fx : uðxÞ ¼ 1g. Thus, we refer to this problem as the nonlocal isoperimetric problem (NLIP), since it is a volume-constrained, nonlocal perturbation of the area functional. Problem (NLIP) is directly related to modeling microphase separation of diblock copolymers (cf. [2], [9], [22], see also the Appendix) and is closely related to models arising in the study of magnetic domains and walls (cf. [11], [17]). Melts of diblock copolymers represent a physical system which exhibits the following phase separation morphology (cf. [6], [34]): the phase separation is periodic on some fixed mesoscopic scale, and within a period cell, the interfaces are close to being area-minimizing. The modeling of (nearly) periodic pattern morphologies via energy minimization involving long and short-range competitions is well-established and ubiquitous in science (cf. [15], [20], [30] and the references therein). Alternatively, problem (NLIP) can be viewed as a simple mathematical paradigm for this type of pattern morphology. Let us be clear on which periodicity we are referring to here. There are two length scales for minimizers of (NLIP). The first is set by the size of the domain W, so that for example in the case W ¼ T n any minimizer can naturally be regarded as periodic with period one. This is not the periodicity we allude to, and the choice of periodic boundary conditions imposed by working on the torus is made for convenience only: Indeed, in Remark 2.8 we indicate how all of our results can be readily adapted to the homogeneous Neumann setting. For g su‰ciently large, a smaller scale is enforced as a weak constraint via interactions between the perimeter and the nonlocal term. Heuristically, it Ðis not hard to see that minimization of the nonlocal term favors oscillation: the constraint u dx ¼ m fixes the measure of the sets fx : uðxÞ ¼ G1g and distributing these two sets in an oscillatory way leads to oscillation of the Laplacian of v through the Poisson equation, thus keeping down the value of the L 2 -norm of ‘v. Accepting that oscillations are preferred, it is not unreasonable to anticipate periodic or nearly periodic structures emerging as the value of g increases. In one space dimension, it has been proven that minimizers of (NLIP) and its di¤use interface version (cf. (4.1) in Appendix) are periodic on a scale determined by g (see [3], [19], [35] for the case m ¼ 0 and [24] for general m). In higher space dimensions, there is evidence to support the contention that minimizers are at least nearly periodic (see [1] and the references therein). Whether or not minimizers are exactly periodic and the precise nature of their geometry within a periodic cell remains an open problem. We expect that minimizers are at least nearly periodic for large g, and this inherent mesoscopic periodicity is one of the reasons why (NLIP) is of interest. Setting g ¼ 0 leads to a fixed volume, area-minimization (isoperimetric) problem (cf. [29]). Posing this problem on the torus gives the periodic isoperimetric problem which, together with its di¤use interface counterpart (the periodic Cahn-Hilliard problem), is the focus of an earlier paper of ours [10]. Note that for these local problems, the imposed periodic boundary conditions associated with working on the torus are crucial—greatly influencing the nature of minimizers. One of the points of this article concerns the observation that in addition to favoring periodicity on a smaller scale, the long-range (nonlocal) term will also a¤ect the geometry of minimizing structures. In fact the phase boundary of minimizers will not, in general, be of constant mean curvature (CMC) as it is for the classical isoperimetric problem where g ¼ 0. This observation is made precise in Theorem 2.3, where a first variation calculation reveals that along the phase boundary of a critical point u one has the condition
Choksi and Sternberg, Nonlocal isoperimetric problem
77
ðn 1ÞHðxÞ þ 4gvðxÞ ¼ l for all x A qfx : uðxÞ ¼ 1g for some constant l. Here HðxÞ denotes the mean curvature of qfx : uðxÞ ¼ 1g. From the criticality condition above, one sees that the phase boundary will be nearly of constant mean curvature precisely when it is close to being a level set of v. Indeed, one of our long term goals is to rigorously address the extent to which minimizers of (NLIP) are close to having CMC interfaces. The connection between CMC surfaces and phase boundaries in diblock copolymer melts is well-established in the literature (see for example [1], [4], [34]). The problem (NLIP) is perhaps the simplest energetic model for phase separation in diblock copolymers which takes into account long-range interactions due to the connectivity of the di¤erent monomer chains. Numerical experiments on the gradient flow for (4.1) also suggest that its minimizers are close to being CMC ([33]). Since in the appropriate regime the G-limit of (4.1) is (NLIP) (cf. [9], [23]), one expects similar behavior for (NLIP). In [27], [26], a spectral analysis of some simple CMC structures (stripes and spots, and rings) is presented to establish instability for su‰ciently large g (so called ‘‘wiggled’’ stripe and spot instabilities) for (4.1). Not surprisingly, this closeness to CMC has also been observed in general reaction-di¤usion-type models for Turing patterns (see [12], [14] and the references therein). In part to address this issue further, a second and technically more involved point of emphasis for this article is the calculation of the second variation of (NLIP). While it is clearly the case that a full understanding of the complex energy landscape for such a nonlocal problem will undoubtedly require tools beyond the second variation, which is inherently local, this calculation seems a reasonable starting point for any stability analysis. Of course the second variation formula for the area functional ðg ¼ 0Þ is a standard calculation leading to the well-known and useful criterion for the stability of constant mean curvature surfaces: Ð
ðj‘qA zj 2 kBqA k 2 z 2 Þ dH n1 ðxÞ f 0
qA
for all smooth functions z satisfying
Ð
zðxÞ dx ¼ 0, cf. e.g. [5], [31]. Here we have let A
qA
denote the set fx : uðxÞ ¼ 1g, ‘qA denotes the gradient relative to the manifold qA and kBqA k 2 denotes the square of the second fundamental form of qA, i.e. the sum of the squares of the principal curvatures. However, in addition to the incorporation of the nonlocal term, we must re-compute the second variation of area since we are taking it about a critical point of Eg , not about a critical point of the pure area functional E0 . We find in our main result, Theorem 2.6, that the stability criterion of non-negative second variation for (NLIP) in the case W ¼ T n takes the form of the following inequality: Ð
ðj‘qA zj 2 kBqA k 2 z 2 Þ dH n1 ðxÞ þ 8g
qA
Ð Ð
Gðx; yÞzðxÞzðyÞ dH n1 ðxÞ dH n1 ðyÞ
qA qA
Ð þ 4g ‘v nz 2 dH n1 ðxÞ f 0; qA
for all smooth functions z : T n ! R satisfying
Ð
zðxÞ dx ¼ 0. Here G denotes the Green’s
qA
function associated with the Laplacian on the torus and n denotes the unit normal to qA
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Choksi and Sternberg, Nonlocal isoperimetric problem
pointing out of A. A slightly modified version also holds on a general bounded domain with homogeneous Neumann boundary conditions; see Remark 2.8. The paper is organized as follows. We carry out the calculation of the first and second variations in Section 2. In Section 3, we present some applications of these formulas, including an analysis of the stability of lamellar structures for small and large values of g (Propositions 3.5 and 3.6). Our hope is that the first and second variation formulas will prove to be useful tools in the stability analysis of critical points of (NLIP) of non-constant mean curvature as well. Finally, in the Appendix we describe the di¤use version of the functional Eg . Acknowledgments. Part of the work for this project was carried out while P.S. was visiting the Institute for Mathematics and its Applications in Minneapolis. He would like to thank the I.M.A. for its hospitality during this visit.
2. The first and second variations of (NLIP) In this section, we calculate both the first and second variation of the nonlocal isoperimetric problem. For the majority of this section, we choose W ¼ T n and work with periodic boundary conditions. This is for convenience only and in Remark 2.8, we address the necessary modifications for working with homogeneous Neumann boundary conditions in a general domain W. To fix notation, we will analyze the functional Eg given by ð2:1Þ
Eg ðuÞ :¼ EðuÞ þ gF ðuÞ;
where E : L 1 ðT n Þ ! R is defined by 8 > < 1 Ð j‘uj EðuÞ ¼ 2 T n > : þy
if juj ¼ 1 a:e:; u A BV ðT n Þ;
Ð
u ¼ m;
Tn
otherwise;
and F : L 1 ðT n Þ ! R denotes the functional
F ðuÞ ¼
8Ð Ð < j‘vj 2 dx if juj ¼ 1 a:e:; u A BV ðT n Þ; u ¼ m; Tn
: þy
Tn
otherwise;
where v : T n ! R1 depends on u as the solution to the problem ð2:2Þ
Dv ¼ ðu mÞ in T n ;
Ð
v ¼ 0:
Tn
Ð In the definition of E, we use j‘uj to denote the total variation measure of u evaluated on Tn T n (cf. [13]). Note in particular that v satisfies periodic boundary conditions on the boundary of ½0; 1 n . We can write v in terms of the Green’s function G ¼ Gðx; yÞ associated with the periodic Poisson problem (2.2). Precisely, for each x A T n , let Gðx; yÞ be the solution to
Choksi and Sternberg, Nonlocal isoperimetric problem
ð2:3Þ
hy Gðx; yÞ ¼ ðdx 1Þ on T n ;
Ð
79
Gðx; yÞ dx ¼ 0;
Tn
where dx is a delta-mass measure supported at x. In particular, if we introduce the fundamental solution 8 1 > > > > > : on ðn 2Þjxj n2
if n ¼ 2; if n > 2;
then, for any fixed x A T n , ð2:4Þ
Gðx; yÞ Fðx yÞ is a C y function ðof yÞ in a neighborhood of x:
The functions G and v are then related by ð2:5Þ
vðxÞ ¼
Ð
Gðx; yÞuðyÞ dy:
Tn
Calculation of the first and second variations of E alone about a critical point of E is a standard procedure (see e.g. [31]) that corresponds simply to the first and second variations of area subject to fixed volume. However, what makes this particular part of our calculation non-standard is that we are not computing this second variation about a critical point of E but rather about a critical point of Eg ; hence the phase boundary will not be of constant mean curvature and the formula will necessarily include an extra term that reflects this change. More significantly, to our knowledge, the variations of F have not been previously computed in this context. Before presenting the result, we should mention that in case the requirement juj ¼ 1 a.e. is dropped, then the calculation of the second variation of F is trivial. To this end, note that ð2:6Þ
F ðuÞ ¼
Ð
Ð Ð j‘vj 2 dx ¼ vðxÞDvðxÞ dx ¼ vðxÞ uðxÞ m dx
Tn
¼
Ð
Tn
vðxÞuðxÞ dx ¼
Tn
Tn
Ð Ð
Gðx; yÞuðxÞuðyÞ dx dy:
Tn Tn
With this representation in hand, one can easily compute d 2 F ðu; u~Þ :¼
Ð Ð d2 F ðu þ e~ uÞ ¼ Gðx; yÞ~ uðxÞ~ uðyÞ dx dy; 2 de je¼0 Tn Tn
Ð where the variations u~ run over all functions in L 1 ðT n Þ satisfying u~ðxÞ dx ¼ 0 so as Tn Ð u ¼ m. However, taking the constraint juj ¼ 1 a.e. into to respect the mass constraint Tn
account puts a severe restriction on the variations. Indeed one should really view Eg as a functional depending on a set, say A H T n , through the formula ð2:7Þ
1 if x A A; uðxÞ ¼ 1 if x A A c :
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Choksi and Sternberg, Nonlocal isoperimetric problem
Here A c denotes the complement of A, i.e. A c ¼ T n nA, and the n-dimensional measure of A, which we write as jAj, is compatible with the mass constraint on u, so that 2jAj 1 ¼ m. Our goal is to compute the first and second variations of Eg as we allow a critical point A H T n to vary smoothly in such a way as to preserve its volume to second order. Given a set A H T n with C 2 boundary, we say a family of sets fAt gt A ðt; tÞ for some t > 0 represents an admissible perturbation (or flow) of A for the purpose of calculating a first variation if the family fAt g H T n satisfies the following three conditions: ð2:8Þ
wAt ! wA
as t ! 0 in L 1 ðT n Þ;
ð2:9Þ
qAt is of class C 2 ;
ð2:10Þ
d jAt j ¼ 0; dtjt¼0
where w denotes the characteristic function of a set. Note that necessarily A0 ¼ A. For the purpose of calculating a second variation, we will need to consider a family of sets fA~t g that in addition to (2.8)–(2.10) satisfies the second order condition d2 jA~t j ¼ 0: dt 2 jt¼0
ð2:11Þ
Corresponding to each of these families, we define Uðx; tÞ and U~ ðx; tÞ respectively by ( 1 if x A At ; 1 if x A A~t ; and U~ ðx; tÞ ¼ ð2:12Þ Uðx; tÞ ¼ c 1 if x A At ; 1 if x A A~tc : Note that Uðx; 0Þ ¼ U~ ðx; 0Þ ¼ uðxÞ given by (2.7). Also, for any t A ðt; tÞ, we define V ð; tÞ and V~ ð; tÞ to be solutions of ð2:13Þ
DV ð; tÞ ¼ Uð; tÞ
Ð
Uðy; tÞ dy;
Tn
ð2:14Þ
DV~ ð; tÞ ¼ U~ ð; tÞ
Ð T
Ð
V ðx; tÞ dx ¼ 0;
Tn
U~ ðy; tÞ dy;
n
Ð T
V~ ðx; tÞ dx ¼ 0;
n
respectively. Note that V ðx; 0Þ ¼ V~ ðx; 0Þ ¼ vðxÞ where v : T n ! R 1 satisfies ð2:15Þ
Dv ¼ ðu mÞ;
Ð
vðxÞ dx ¼ 0;
Tn
with u given by (2.7). We now state precisely our notion of criticality and stability. Definition 2.1. A function u given by (2.7) is a critical point of Eg if ð2:16Þ
d Eg Uð; tÞ ¼ 0 dtjt¼0
for every U ¼ Uðx; tÞ associated with an admissible family fAt g satisfying (2.8)–(2.10).
Choksi and Sternberg, Nonlocal isoperimetric problem
81
Definition 2.2. A critical point u of Eg is stable if d2 Eg U~ ð; tÞ f 0 2 dt jt¼0
ð2:17Þ
for every U~ ¼ U~ ðx; tÞ associated with an admissible family fA~t g satisfying (2.8)–(2.11). Clearly if u is a local minimizer in L 1 of Eg such that qA is C 2 then in particular u will be stable in this sense. For the remainder of the paper, we use H n1 to denote ðn 1Þ-dimensional Hausdor¤ measure. Our two main results are the following: Theorem 2.3. Let u be a critical point of Eg given by (2.7) such that qA is C 2 with mean curvature H : qA ! R. Let z be any smooth function on qA satisfying the condition Ð zðxÞ dH n1 ðxÞ ¼ 0: qA
Then for v solving (2.2) one has the condition Ð ð2:18Þ ðn 1ÞHðxÞ þ 4gvðxÞ zðxÞ dH n1 ðxÞ ¼ 0: qA
Hence there exists a constant l such that ð2:19Þ
ðn 1ÞHðxÞ þ 4gvðxÞ ¼ l for all x A qA:
Remark 2.4. When g ¼ 0, one is simply studying critical points of area subject to a volume constraint so, as is well-known, one gets constant mean curvature as a condition of criticality. For g > 0 but small one sees from (2.1) that the curvature is almost constant, but it will not actually be constant unless it happens that qA is a level set of v solving (2.2). This will be the case for a periodic lamellar structure (see Proposition 3.3) but note, for example, that a sphere (or periodic array of spheres) will never be a critical point—unless one works, as in [26], within the ansatz of radial symmetry. Remark 2.5. A comment needs to be made regarding the assumed regularity of the critical point in Theorem 2.3, as well as that to be made in Theorem 2.6 to follow. When g ¼ 0, it is well-known (see for example [16]) that in dimensions n < 8, the phase boundary associated with any L 1 -local minimizer must have constant mean curvature and be an analytic ðn 1Þ-dimensional manifold, while in dimensions n f 8, the same is true o¤ of a (perhaps empty) singular set of Hausdor¤ dimension at most n 8. While the phase boundary associated with a local minimizer for g > 0 will, in general, no longer have constant mean curvature, we strongly suspect that this lower-order, compact perturbation will not destroy regularity, so we expect that the phase boundary associated with a local minimizer of Eg is still an analytic ðn 1Þ-dimensional manifold in dimensions n < 8, and in particular is C 2 . In any event, the hypothesis of a critical point with C 2 boundary is henceforth adopted, though accomodation could be made for a low dimensional singular set as was done in [32]. Computation of a second variation about a critical point u then leads to:
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Choksi and Sternberg, Nonlocal isoperimetric problem
Theorem 2.6. Let u be a stable critical point of Eg given by (2.7) such that qA is C 2 . Let z be any smooth function on qA satisfying the condition Ð zðxÞ dH n1 ðxÞ ¼ 0: qA
Then for v solving (2.2) one has the condition ð2:20Þ
JðzÞ :¼
Ð
ðj‘qA zj 2 kBqA k 2 z 2 Þ dH n1 ðxÞ
qA
þ 8g
Ð Ð
Gðx; yÞzðxÞzðyÞ dH n1 ðxÞ dH n1 ðyÞ
qA qA
Ð þ 4g ‘v nz 2 dH n1 ðxÞ f 0: qA
Here ‘qA z denotes the gradient of z relative to the manifold qA, BqA denotes the second n1 P 2 fundamental form of qA so that kBqA k 2 ¼ ki where k1 ; . . . ; kn1 are the principal curvai¼1
tures and n denotes the unit normal to qA pointing out of A. A formal derivation of essentially the same formula appears in the appendix of [20]. Remark 2.7. We remark that for any z, Ð Ð Gðx; yÞzðxÞzðyÞ dH n1 ðxÞ dH n1 ðyÞ f 0; qA qA
and hence this term is stabilizing. To see this, let m denote the measure given by zH n1 K qA. Since m so defined obviously satisfies m f H n1 K qA, it is easy to check that the potential w given by Ð wðxÞ ¼ Gðx; yÞ dmðyÞ Tn
is bounded. Then it follows from classical potential theory that in fact w is an H 1 (weak) solution to the equation ð2:21Þ
Dw ¼ m on T n ;
and satisfies Ð Ð qA qA
Gðx; yÞzðxÞzðyÞ dH n1 ðxÞ dH n1 ðyÞ ¼
Ð Ð Tn Tn
Gðx; yÞ dmðxÞ dmðyÞ ¼
Ð
j‘wj 2 dx
Tn
(cf. [18], Chapter 1). Alternatively, one may view the above as kmk2H 1 ðT n Þ , a version of the H 1 -norm squared of the measure m (cf. [28]). Before beginning the proofs of our two theorems, we indicate a way to construct admissible families (flows) fAt g and fA~t g satisfying (2.8)–(2.10) and (2.8)–(2.11) respectively. Our approach here is rather explicit and is therefore e‰cient for pursuing our goals. We use an ODE to produce a flow deformation of A whose boundary instantaneously moves according to a perturbation vector field X , and which instantaneously preserves volume (in
Choksi and Sternberg, Nonlocal isoperimetric problem
83
the sense of (2.10)). We then apply a correction to insure it instantaneously preserves volume to second order as well (i.e. (2.11) is satisfied). This is essentially the approach of [32] where, in fact, a flow was obtained which identically preserved volume for a finite time interval. However, for the purposes there and here, the instantaneous notions of (2.10) and (2.11) su‰ce. We should note that if one only wanted to establish the existence of a volume preserving flow, instantaneously associated with a perturbation vector field X , a simple application of the implicit function theorem, as was done in [5], would su‰ce. However, this construction would not make the computations necessary for the second variation explicit. In any event, our hands-on approach via the correction A~t does not significantly expand the proofs. Let X : T n ! R n be a C 2 vector field such that Ð ð2:22Þ X n dH n1 ðxÞ ¼ 0; qA
where n denotes the outer unit normal to qA. Then let C : T n ðt; tÞ ! T n solve qC ¼ X ðCÞ; qt
ð2:23Þ
Cðx; 0Þ ¼ x;
for some t > 0 and define ð2:24Þ
At :¼ CðA; tÞ:
Expanding in t we find that ð2:25Þ
where Z :¼ ð2:26Þ
1 DCð; tÞ ¼ I þ t‘X þ t 2 ‘Z þ oðt 2 Þ; 2 q2C has i th component given by qt 2 jt¼0 Z ðiÞ ¼ XxðiÞj X ð jÞ :
Here and throughout, we invoke the summation convention on repeated indices. Letting J C denote the Jacobian of C, we then invoke the matrix identity ð2:27Þ
1 det I þ tA þ t 2 B þ oðt 2 Þ 2 1 ¼ 1 þ t trace A þ t 2 ½trace B þ ðtrace AÞ 2 traceðA 2 Þ þ oðt 2 Þ; 2
which holds for any square matrices A and B. This yields ð2:28Þ
q J C ¼ trace ‘X ¼ div X : qtjt¼0
Consequently, (2.22) and the Divergence Theorem imply that
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Choksi and Sternberg, Nonlocal isoperimetric problem
Ð d d jAt j ¼ J C dx ¼ 0; dtjt¼0 dtjt¼0 A
ð2:29Þ
and the family of sets fAt g constitutes an admissible perturbation of A for the purpose of calculating a first variation, cf. (2.10). However, we note that further use of (2.25) and (2.27) yields Ð q2J C d2 jA j ¼ dx t 2 dt 2 jt¼0 A qt jt¼0 Ð ¼ div Z þ ðdiv X Þ 2 XxðiÞj XxðijÞ dx
ð2:30Þ
A
¼
Ð div X ðxÞ X ðxÞ n dH n1 ðxÞ; qA
where we obtained the last line through the two identities ð2:31Þ
div Z ¼ XxðiÞi xj X ð jÞ þ XxðiÞj XxðijÞ
ð2:32Þ
div ðdiv X ÞX ¼ ðdiv X Þ 2 þ XxðiÞi xj X ð jÞ :
and
Hence, in light of the requirement (2.11), we see through (2.30) that fAt g is not, in general, admissible when calculating a second variation, and we must modify it. d2 jAt j 3 0, then we define the modified perturbation fA~t g as dt 2 jt¼0 follows. We first introduce the distance function d : T n ðt; tÞ ! R by If it turns out that
dðx; tÞ ¼ distðx; qAt Þ;
ð2:33Þ
which will be smooth on ½0; tÞ and ðt; 0 for small enough t in light of the assumed regularity of qA. We will make frequent use of the fact that j‘dðx; tÞj ¼ 1. We next note that for At constructed as above, the quantity jAt j is a C 2 function of t. Hence by (2.29) and (2.30), there must be a t 0 > 0 such that for jtj < t 0 either jAt j < jAj or jAt j > jAj. If jAt j < jAj, then for jtj < t 0 we let A~t :¼ At W fx A Atc : dðx; tÞ < c1 t 2 g;
ð2:34Þ while if jAt j > jAj, we let
A~t :¼ At nfx A At : dðx; tÞ < c1 t 2 g:
ð2:35Þ
Here the constant c1 is given by ð2:36Þ where the average
c1 ¼ Ð qA
1Ð div X ðxÞ X ðxÞ n dH n1 ðxÞ; 2 qA
denotes the integral divided by H n1 ðqAÞ.
Choksi and Sternberg, Nonlocal isoperimetric problem
85
We will verify that condition (2.11) holds in the case jAt j < jAj. The other case is handled similarly. Writing jA~t j ¼ jAt j þ ðjA~t j jAt jÞ we use the co-area formula (cf. [13] or [31]) to see that 2 Ð d2 ~t j jAt jÞ ¼ d ðj A j‘dðx; tÞj dx 2 2 dt jt¼0 dt jt¼0 A c Xf0 0, u is a critical point of Eg if and only if for i ¼ 0; 1; . . . ; N 1, ða2iþ1 a2i Þ ¼
mþ1 ; 2N
ða2iþ2 a2iþ1 Þ ¼
1m : 2N
Proof. Let S :¼ fx A T n j uðxÞ ¼ 1g. Since HðxÞ ¼ 0 for all x A qS, Theorem 2.3 implies that if u is a critical point, then qS must be a level set of the associated v solving (2.2). Clearly, vðxÞ ¼ vðx1 Þ and without loss of generality, we may assume vð0Þ ¼ 0. Hence we require vðai Þ ¼ 0 for all i ¼ 0; . . . ; 2N. Since v 00 ðx1 Þ ¼ f ðx1 Þ m
on T 1 ;
v consists of piecewise parabolas defined on the subinterval ðai ; aiþ1 Þ which alternate pointing upwards (with convexity 1 m) and pointing downwards (with concavity 1 m). The conditions that vðai Þ ¼ 0 for each i, and the fact that v must be C 1 at each ai directly imply that the parabolas must repeat themselves every second step. This is illustrated in Figure 1. r
Choksi and Sternberg, Nonlocal isoperimetric problem
101
Figure 1. A function v corresponding to a lamellar critical point.
Remark 3.4. We point out that the conditions for criticality of strips described in Proposition 3.3 are independent of g. Thus, a lamellar critical point for one value of g is automatically a critical point for all values of g. We presume that this is the only example of a subset of T n that is critical for two di¤erent g values. Observe from the criticality condition (2.19) that any such subset would necessarily have to possess a boundary of constant mean curvature with that boundary being a level set of the associated v solving Poisson’s equation. We now examine the stability of lamellar critical points of Eg . In particular, we study how the value of g a¤ects stability. One certainly expects that as g increases, stability will require a smaller length scale for periodic structures. We will focus on the case of a single strip S. We will argue that S is stable for small g and unstable for g su‰ciently large. For simplicity of presentation only, we will work in two space dimensions. Both the stability and the instability result below readily generalize to strips in higher dimensions with only minor changes. For 0 < b < a < 1, let S H T 2 denote the set ½b; a ½0; 1. Then qS ¼ ðfbg ½0; 1Þ W ðfag ½0; 1Þ: Also note that the function u associated with S via (2.7) is given by 1 for x1 A ðb; aÞ; uðx1 ; x2 Þ ¼ 1 for x1 B ðb; aÞ: Proposition 3.5. Let S H T 2 be a horizontal or vertical strip. Then there exists a value g0 such that S is a stable critical point of Eg for all positive g < g0 . Proposition 3.6. For g su‰ciently large, S is unstable. We first prove Proposition 3.5. Proof. We first note that since u ¼ uðx1 Þ, one has v ¼ vðx1 Þ satisfying v 00 ¼ u m A L p for all p > 2. It then follows from standard regularity theory that v A W 2; p for all p > 2, and in particular that v A C 1 . If we write qS ¼ G1 W G2 where G1 ¼ fbg ½0; 1 and G2 ¼ fag ½0; 1, then the first component of the outer unit normal to S, which we denote here by z0 , is given by
102
Choksi and Sternberg, Nonlocal isoperimetric problem
1 for x A G1 ; z0 ðxÞ ¼ þ1 for x A G2 : We can then invoke Proposition 3.1 and (3.3) to see that 0 ¼ Jðz0 Þ ¼ 8g
ð3:5Þ
Ð Ð
Ð Gðx; yÞz0 ðxÞz0 ðyÞ dH 1 ðxÞ dH 1 ðyÞ þ 4g ‘v n dH 1 ðxÞ;
qS qS
qS
since of course, BqA 1 0 for the strip. Now let z : qS ! R be any smooth function satisfying Ð
ð3:6Þ
z dH 1 ðxÞ ¼ 0:
qS
We will write z as z ¼ f cz0 where c :¼
Ð
zðxÞ dH 1 ðxÞ and f is given by
G1
ð3:7Þ
f ðxÞ ¼
z c for x A G1 ; z þ c for x A G2 :
Denoting f1 :¼ f K G1 and f2 :¼ f K G2 we observe that
Ð1
fi ðx2 Þ dx2 ¼ 0 for i ¼ 1; 2. We
0
will argue that JðzÞ > 0 for g su‰ciently small, provided f E 0, that is, provided z does not correspond to a translation. To this end, we compute
ð3:8Þ
Ð1 Ð1 JðzÞ ¼ ð f10 Þ 2 dx2 þ ð f20 Þ 2 dx2 0
0
þ 8g
Ð Ð
Ð Gðx; yÞ f ðxÞ f ðyÞ dH 1 ðxÞ dH 1 ðyÞ þ 4g ‘v nf 2 dH 1 ðxÞ
qS qS
qS
Ð Ð Ð Gðx; yÞz0 ðxÞz0 ðyÞ dx dy þ 4g ‘v n dH 1 ðxÞ þ c 2 8g qS qS
qS
Ð Ð þ 8gc Gðx; yÞ f ðxÞz0 ðyÞ dH 1 ðxÞ dH 1 ðyÞ qS qS
þ
Ð Ð
Gðx; yÞz0 ðxÞ f ðyÞ dH 1 ðxÞ dH 1 ðyÞ
qS qS
Ð
þ 8gc ‘v nf z0 dH 1 ðxÞ qS
ð3:9Þ
fp
2
Ð
f 2 dH 1 ðxÞ þ 8g
qS
Ð Ð
qS qS
Ð
þ 4g ‘v nf 2 dH 1 ðxÞ qS
Gðx; yÞ f ðxÞ f ðyÞ dH 1 ðxÞ dH 1 ðyÞ
103
Choksi and Sternberg, Nonlocal isoperimetric problem
Ð Ð þ 8gc Gðx; yÞ f ðxÞz0 ðyÞ dH 1 ðxÞ dH 1 ðyÞ qS qS
þ
Ð Ð
Gðx; yÞz0 ðxÞ f ðyÞ dH ðxÞ dH ðyÞ 1
1
qS qS
Ð
þ 8gc ‘v nf z0 dH 1 ðxÞ; qS
through (3.5) and the Poincare´ inequality. Now Ð
Ð1 Ð1 ‘v nf z0 dx ¼ vx1 ðaÞ f2 ðx2 Þ dx2 þ vx1 ðbÞ f1 ðx2 Þ dx2 ¼ 0 0
qS
since
Ð1
0
fi ðx2 Þ dx2 ¼ 0, and so the last term in (3.9) vanishes. Also, note that the function
0
wðyÞ :¼
Ð
Gðx; yÞz0 ðxÞ dH 1 ðxÞ
qS
is a (weak) solution to the problem Dwðx1 ; x2 Þ ¼ mz0 where mz0 ¼ mz0 ðx1 Þ is the measure given by mz0 ¼ 1H 1 K fðx1 ; x2 Þ : x1 ¼ ag 1H 1 K fðx1 ; x2 Þ : x1 ¼ bg: Hence, in particular, w ¼ wðx1 Þ which means that Ð Ð
Gðx; yÞz0 ðxÞ f ðyÞ dH 1 ðxÞ dH 1 ðyÞ ¼
qS qS
Ð
wðyÞ f ðyÞ dH 1 ðyÞ
qS
Ð1 Ð1 ¼ wðaÞ f2 ðx2 Þ dx2 þ wðbÞ f1 ðx2 Þ dx2 ¼ 0: 0
Therefore, the second to last line of (3.9) vanishes as well. Since Ð Ð
Gðx; yÞ f ðxÞ f ðyÞ dH 1 ðxÞ dH 1 ðyÞ f 0;
qS qS
we arrive at the inequality JðzÞ f ðp 2 4gkv 0 kLy Þ
Ð
f 2 ðxÞ dH 1 ðxÞ:
qS
p2 , we have the desired result. r Thus, choosing g0 ¼ 4kv 0 kLy We now prove Proposition 3.6.
0
104
Choksi and Sternberg, Nonlocal isoperimetric problem
Proof. Note that by the divergence theorem, Ð
ð3:10Þ
Ð ‘vðy1 ; y2 Þ n dy2 ¼ hvðyÞ dy S
f bg½0; 1Wfag½0; 1
Ð ¼ m uðyÞ dy S
¼ ðm 1ÞjSj < 0: Hence, setting gðy2 Þ :¼ ‘vða; y2 Þ n; we may assume without loss of generality that Ð1
ð3:11Þ
gðy2 Þ dy2 ¼: c1 < 0:
0
Let k > 1 be a positive integer to be chosen shortly, and consider the function zk set to be identically zero on fbg ½0; 1 and set to equal sinð2pky2 Þ on fag ½0; 1. There are three non-zero terms in Jðzk Þ of (2.20) since BqS 1 0. The first involving ‘qA zk is readily calculated to equal 2p 2 k 2 . The next non-zero term can be written as g times
ð3:12Þ
Ð1 Ð1 8 G ða; x2 Þ; ða; y2 Þ sinð2pkx2 Þ dx2 sinð2pky2 Þ dy2 : 0 0
We claim that (3.12) can be made arbitrarily small by choosing k su‰ciently large. To this end, denote the function in the square parenthesis of (3.12) by fk ðy2 Þ. Note that (cf. (2.4)) ð3:13Þ
Ð1 G ða; x2 Þ; ða; y2 Þ dx2 < y for every y2 A ½0; 1Þ; 0
and further Ð1 Ð1 G ða; x2 Þ; ða; y2 Þ dx2 dy2 < y:
ð3:14Þ
0 0
By (3.13) and the Riemann-Lebesgue Lemma, for each y2 we have fk ðy2 Þ ! 0 as k ! y: Hence by (3.14) and the Lebesgue Dominated Convergence Theorem,
ð3:15Þ
Ð1 0
fk ðy2 Þ sinð2pky2 Þ dy2 ! 0 as k ! y:
Choksi and Sternberg, Nonlocal isoperimetric problem
105
The third and last remaining non-zero term in (2.20) takes the form Ð1 Ð1 1 cosð4pky2 Þ 4g gðy2 Þ sin 2 ð2pky2 Þ dy2 ¼ 4g gðy2 Þ dy2 ; 2 0 0 and hence by (3.11) and the Riemann-Lebesgue Lemma, we have ð3:16Þ
lim
Ð1
k!y 0
gðy2 Þ sin 2 ð2pky2 Þ dy2 ¼
c1 < 0: 2
Therefore, we can choose k su‰ciently large such that for any g > 0 the sum of the last two terms of (2.20) is, say, less than gc1 =4. Fixing such a k, we take g0 to be the smallest value of g such that gjc1 j=4 > 2p 2 k 2 . Hence Jðzk Þ < 0 and the stability criterion (2.20) is violated. r Remark 3.7. Proposition 3.6 can easily be extended to the case of N strips. In this case, consider S and the associated u as defined in Proposition 3.3. Then following (3.10), one finds that for some y1 ¼ aj , Ð fy1 ¼aj g
‘vðaj ; y2 Þ n
0, consider
ð4:1Þ
E; g
8 > 1 ð1 u 2 Þ 2 1Ð : þy otherwise:
Here W is again either the torus T n or a general domain with smooth boundary, and v is related to u and m via hv ¼ u m
on W;
Ð W
vðxÞ dx ¼ 0;
Choksi and Sternberg, Nonlocal isoperimetric problem
107
with the di¤erential equation solved on T n in the former case, and with homogeneous Neumann boundary conditions in the latter. This functional is essentially the energy first derived by Ohta and Kawasaki [22] to model microphase separation of diblock copolymers. As written in its non-dimensional, appropriately re-scaled form, it may be viewed as a nonlocal Cahn-Hilliard type functional (cf. [7], [8], [9], [21], [23], [25]). A full derivation can be found in [9]. Note that the third term in (4.1) represents a compact perturbation with respect to the basic L 2 (or L 1 ) topology. Hence it easily follows (cf. [23]) from the definition of G-convergence that the G-limit problem (as ! 0 and in the L 1 -topology) is equivalent to (NLIP).
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Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada e-mail:
[email protected] Department of Mathematics, Indiana University, Bloomington, IN, USA e-mail:
[email protected] Eingegangen 27. Februar 2006, in revidierter Fassung 11. Juli 2006
